Summary: A representation theorem for stochastic
processes with separable covariance functions,
and its implications for emulation
Jonathan Rougier
Department of Mathematics
University of Bristol
Abstract
Many applications require stochastic processes specified on two- or higher-
dimensional domains; spatial or spatial-temporal modelling, for exam-
ple. In these applications it is attractive, for conceptual simplicity and
computational tractability, to propose a covariance function that is sep-
arable; e.g. the product of a covariance function in space and one in time.
This paper presents a representation theorem for such a proposal, and
shows that all processes with continuous separable covariance functions
are second-order identical to the product of second-order uncorrelated
processes. It discusses the implications of separable or nearly separable
prior covariances for the statistical emulation of complicated functions
such as computer codes, and critically reexamines the conventional wis-
dom concerning emulator structure, and size of design.
Keywords: Stochastic process, spatial-temporal modelling, kth-order